Describing motion from position-time graphs

 

The position-time graph to the right represents the motion of a cart on a track.  The motion detector points to the left.  Explain how to reproduce this graph.

When describing motion, students only need to address two elements: 

a. Which way is the cart moving?

b. Is the cart speeding up or slowing down?

Yet, students trying to answer this question will tie themselves in knots addressing how the cart is "accelerating", trying to use words like displacement, velocity, vector, negative, positive, etc.

Don't even give your class the string with which to tie knots.  Get them in the habit of addressing each of these two questions only; and get them starting every response with a fact of physics.

For the initial foray into position-time graphs, only three facts are relevant: 

1. A position-time slope like a front slash / means the object is moving away from the detector.
2. A position-time slope like a back slash \ means the object is moving toward the detector.
3. To determine how fast an object is moving, look at the steepness of the position-time graph.

a. Which way is the cart moving?  Do NOT accept an answer that begins with "The cart is moving right."  I don't care what else this response says, it is wrong on its face.  A response must begin with a fact of physics.*

*"But that's not fair, Greg! A student who says "the cart is moving west because the slope is negative" is right!"  See, I'm not concerned with whether this particular answer is right or wrong - I'm concerned with my students developing a long-term deep understanding of position-time graphs, such that they can handle any question on a high-stakes exam as easily as Serena Williams handles a shoulder-high volley at the net.  Imaging Serena as a wee lass asking someone to hit her volleys for practice... and that someone kept lobbing her.  "Please hit me volleys."  "But I won the point!" says her suddenly FORMER practice partner.

"A position-time slope like a back slash \ means the object is moving toward the detector.  This graph is always a back slash, so the cart moves opposite the way the detector is pointing - the cart moves RIGHT."

b. Is the cart speeding up or slowing down?  Similarly, do not accept any answer that doesn't start with a fact of physics - especially do not accept an answer that references acceleration.  Yes, it is technically possible for an experienced physicist with a deep understanding of mathematical physics to recognize that the concave-down graph means the second time derivative is negative, and the negative slope means negative velocity, and to connect that to the magnitude of the velocity vector getting larger.  Aarrgh!  No!  No introductory physics student thinks this way!*

* And the vanishingly rare unicorn who can in fact think this way should have no trouble whatsoever using the much simpler facts above to reason through this graph.  So make the unicorn do so.

"To determine how fast an object is moving, look at the steepness of the position-time graph.  This graph is always getting steeper, so the cart always speeds up."

c. Now go reproduce the graph with a motion detector and cart on a track.  Amazingly, even after answering these two questions correctly, about 20% of the class will still set up the situation incorrectly.  They'll claim they need a curved track.  They'll have the cart moving away from the detector because "the graph is sloped down", even though they just wrote clearly on their own paper that the cart moves toward the detector!

Let them mess up.  

When the students have trouble reproducing this graph, they're confronting their personal misconceptions.  Even if a friend just shows them what to do, they'll see for themselves, "oh, the cart had to move toward the detector and speed up!  Now I get it!"  Or, you can ask them to read back to you what they wrote: "Which way is the cart moving, again?  And is the cart speeding up or slowing down?  So did you set up the cart moving toward the detector and speeding up?"

I don't like having students use their hands or their bodies to reproduce motion.  You'll end up in arguments about whether the graph does or doesn't look like it's supposed to, because it's really tough to keep a hand or a whole body continually speeding up for a second or two.  No, use a cart on a track!  This can be done with a fan cart on a flat track, or with a free-wheeling cart on a slanted track.  Insist on seeing one full second of motion - then you won't see just the 0.1 s when the student pushed the cart to get it started on the incorrect motion, and you won't have to argue about why that's wrong.  It's pretty much impossible to do this wrong and get a good-looking graph that's at least one second long.